We assume an input covariance constraint \( \operatorname{cov}(X) \preceq S \)  for some covariance matrix \(S \succeq 0 \) . The capacity \(\mathcal {C}(S)\)  is the closure of all achievable rate triples. Since the capacity region is convex set, we can characterize the capacity by its supporting hyperplanes. Equivalently, it can be expressed by solving the following optimization problemWe omit the case that \(R_{0}=0\) , because it degenerates to the MIMO Gaussian broadcast channel with only private messages and has been resolved in [1]}, [2]}.
\( \max _{(R_{0},R_{1}, R_{2}) \in \mathcal {C}(S)} R_{0} + \mu _{1}R_{1} + \mu _{2}R_{2}.\) 
